Mode-matching without root-finding:application to a dissipative silencer∗

Jane B. Lawrie†

School of Information Systems, Computing and Mathematics

Mathematical Sciences, Brunel University, Uxbridge UB8 3PH, UK

jane.lawrie@brunel.ac.uk

Ray Kirby

School of Engineering and Design

Mechanical Engineering, Brunel University, Uxbridge UB8 3PH, UK

ray.kirby@brunel.ac.uk

21st February, 2006

Abstract

This article presents an analytic mode-matching approach suitable for modelling the prop-

agation of sound in a two-dimensional, three-part, ducting system. The approach avoids

the need to the find roots of the characteristic equation for the middle section of the duct

(the component) and is readily applicable to a broad class of problems. It is demonstrated

that the system of equations, derived via analytic mode-matching, exhibits certain fea-

tures which ensure that they can be re-cast into a form that is independent of the roots of

the characteristic equation for the component. The precise details of the component are

irrelevant to the procedure; it is required only that there exists an orthogonality relation,

or similar, for the eigenmodes corresponding to the propagating wave-forms in this region.

The method is applied here to a simple problem involving acoustic transmission through a

dissipative silencer of the type commonly found in heating ventilation and air-conditioning

(HVAC) ducts. With reference to this example, the silencer transmission loss is computed,

and the power balance for the silencer is investigated and is shown to be an identity that

is necessarily satisfied by the system of equations, regardless of the level of truncation.

∗Portions of this work were presented in “On analysing a dissipative silencer: a mode-matching ap-

proach” Proceedings of IUTAM 2002/04, Liverpool, U.K. July 2002†Corresponding author.

1

I Introduction

Guided waves can be observed in a wide range of physical situations. Examples include

water waves propagating along a channel, seismological waves travelling through a layer

of rock, the transmission of electromagnetic waves along an optical fibre and acoustical

waves manifesting as noise in heating ventilation and air conditioning (HVAC) ducts. Very

often the physical problems of interest can be formulated as boundary value problems that

are amenable to analytic solution methods such as the Wiener-Hopf technique or mode-

matching. A precursor to these methods, however, is accurate knowledge of sufficient

admissible wavenumbers. These are usually defined in terms of a characteristic equation,

the complexity of which depends on both the equations governing the waveguide media

and on the boundary conditions. Only in the simplest of problems can the wavenumbers be

expressed in closed form and for problems involving elastic or porous media and/or flexible

walled ducts numerical “root-finding” is a normal procedure. Root-finding, whilst tedious,

is by no means an impossible task but the difficulties should not be underestimated.

Of course, the Wiener-Hopf and mode-matching techniques are not necessarily the only

viable analytic solution methods. A wide range of other techniques are available including

those based on Green’s theorem and a variety of Fourier integral approaches. Some of

these approaches successfully avoid the process of root-finding, see for example Huang;4,5

however, such methods are not always easily generalised.

In the following discussion attention is restricted to two-dimensional waveguides. As-

sociated with every mode is an axial and at least one transverse wavenumber which are

related through the governing equation(s). Further, it is assumed that the waveguide

boundary conditions contain only even derivatives in the axial direction.10 Then, broadly

speaking, the characteristic equation, which will be meromorphic and expressed in terms

of the transverse wavenumber, can be broken down into four categories. First, there is

the simple trigonometric form, such as that for a rigid walled acoustic duct, for which the

roots can be written down in closed form. Second, there are those that involve a linear

combination of trigonometric terms and possibly algebraic terms but which have only real

parameters and involve only even powers of at most one transverse wavenumber. For

such characteristic functions the roots always lie on either the real axis or the imaginary

axis. Furthermore, the asymptotic form of the roots is usually known. The characteristic

equations for flexible-walled acoustic ducts10 are typical of this class. The third class of

2

characteristic function is much the same as the second other than it will involve even

powers of two related transverse wavenumbers. Waveguides comprising slabs of an elastic

material typically have this type of characteristic function. In this case the roots may be

real, imaginary or complex yet much may be known about their location1 and asymptotic

form. The fourth class of characteristic equation has all the properties of those in class

three but, in addition, the physical parameters are complex. This class of characteristic

equation arises in the study of dissipative silencers in which the absorbent material is

modelled as an equivalent fluid.

The latter class of characteristic equation is known to be problematic when it comes

to root-finding. This is because the roots do not necessarily lie on or close to any specified

curve in the complex plane which makes it difficult to choose appropriate initial guesses for

the root-finding algorithm. Further, the complex parameters of the dispersion relation are

usually frequency dependent and thus the location of the roots can vary, often significantly

and in arbitrary direction, with small variations in frequency. If one desires to find the

roots at one specific frequency then usually, with perseverance, this can be done. Then,

if one wishes to plot a given physical property against frequency it is usual to adopt

a tracking approach. That is, the roots successfully located for frequency f0 are used

as initial guesses for frequency f0 + ε, ε << 1. Although this approach increases the

likelihood of finding all required roots as frequency varies, it can fail even for frequency

shifts of ε < 0.01 Hz. Not only is it difficult to locate all the roots in a specified region

of the complex plane at all desired frequencies, but there are very few reliable techniques

by which to determine whether all the roots have been found. The Argument Principle

is the only rigorous test, but if one or more roots are found to be missing it is of limited

help in locating them. Traditional algorithms used for solving this class of characteristic

equation include the Newton-Raphson method and variations such as the secant13 and

Muller’s2 methods. None of these are robust in the sense that modes are easily missed if

the initial guesses are not sufficiently close to the actual root. Although the effect of a

missing root on the results depends on the location of the root, it is always undesirable

and often the cause of significant inaccuracies.7

Analytic solution methods that avoid root-finding are highly desirable but few and

far between. Huang,4,5 for example, considers a class of problem whereby a membrane

is inserted into a finite section, say −` ≤ x ≤ `, of an otherwise infinite rigid duct. He

3

employs a technique whereby the sound field within the duct is represented by an infinite

sum of Fourier integrals, each one forced by a velocity distribution sin[nπ(x + `)/(2`)] on

the duct surface for −` ≤ x ≤ `. The Fourier coefficients of the pressure field are then

determined by substituting an evaluated form of the Fourier integrals into the membrane

condition. This works well for the class of problem that Huang considers but the method

is limited to situations in which the membrane forms part of the duct surface. The

method is not applicable to situations whereby there is a structure, such as a finite length

membrane, positioned parallel to the duct wall within the fluid region or to situations in

which part of the fluid region is layered comprising, for example, alternate layers of fluid

and porous material. Further, the extension of Huang’s approach to the the situation in

which the membrane is replaced by an elastic plate is straightforward only for the case

in which the plate edges are pin-jointed. There is a clear need for analytic techniques

that can tackle a wide range of such geometries and do not depend on the roots of the

characteristic equation.

This article offers an approach by which root-finding can be avoided for a broad class

of problem: the propagation of sound in a two-dimensional, three-part ducting system

see, for example, figure 1. The class of problem is described in section II and a general

mode-matching solution is presented. This solution exhibits certain features that ensure

it can always be recast, using a contour integral technique, into a form that is independent

of the roots of the characteristic equation for the middle region. This approach completely

by-passes the need to solve the characteristic equation, and provides robust and accurate

expressions by which to compute any physical quantity of interest. In section III attention

is focussed on a particular example: the propagation of sound through a simple silencer

comprising a finite length of lined duct. The contour integral technique is demonstrated

and a detailed analysis of the power balance is presented. Further analysis proves that

the power balance is an identity which, regardless of the severity of the truncation, is

automatically satisfied by the system of equations relating the reflection and transmission

coefficients. The latter result is well established for non-dissipative systems9,14 but, to

the authors’ knowledge, has not previously been proven for a dissipative system. Nu-

merical results, in terms of transmission loss and power absorbed across each surface of

the liner, are presented for two different silencer configurations in section IV. In section

V a discussion of the potential extensions and limitations of this “root-free” approach is

4

presented.

II The general problem

In this section the general problem of determining the sound field in a two-dimensional,

three-part ducting system is discussed. It is demonstrated that, for the class of problem

to be considered, the system of equations derived via mode-matching have a particular

structure that ensures that they can always be re-cast into a form that is independent of

the roots of the characteristic equation. The system comprises inlet and outlet ducts lying

in the regions 0 ≤ y ≤ b, x < 0 and 0 ≤ y ≤ b, x > 2¯ of a Cartesian frame of reference. A

“silencer-type” component is sandwiched between them occupying the region 0 ≤ y ≤ h,

0 ≤ x ≤ 2¯ where h may be greater than, less than or equal to b. The vertical line

segments joining the points y = h to y = b at x = 0 and x = 2` (when h 6= b) are assumed

to be rigid. Thus, the duct geometry is symmetrical about the vertical line x = `. Figure

1 shows a specific example of a typical three-part ducting system in which h = b. The

analysis that follows is, however, in no way restricted to this case. A compressible fluid

of sound speed cf and density ρ fills the interior of the duct and region exterior to the

duct is in vacuo. The incident sound field is assumed to have harmonic time dependence,

e+iωt, where ω is related to the frequency by ω = 2πf , and propagates through the fluid

in the positive x direction towards x = 0.

It is convenient to non-dimensionalise the boundary value problem using typical length

and time scales k−1 and ω−1 where ω = cfk. Thus, x = kx, y = ky etc. where the “barred”

quantities are dimensional. The time-independent fluid velocity potentials for each duct

region are φ1(x, y), x < 0; φ2(x, y), 0 < x < 2` and φ3(x, y), x > 2` (see figure 1).

For the inlet and outlet ducts (x < 0 and x > 2` respectively) the velocity potentials

are governed by Helmholtz’s equation and must satisfy specified boundary conditions at

y = 0 and y = b. Thus,

φ1 =∞∑

j=0

FjZj(y)e−ηjx +∞∑

j=0

AjZj(y)eηjx; (1)

φ3 =∞∑

j=0

DjZj(y)e−ηj(x−2`) (2)

where Aj, Dj, j = 1, 2, 3 . . . are the complex amplitudes of the reflected and transmitted

modes and ηj is the axial wavenumber of the duct modes. The incident sound field in

5

(1) accommodates multi-modal forcing for which the modal amplitudes Fn will inevitably

depend on the characteristics of the noise source (typically a fan). The eigen-modes Zj(y),

j = 0, 1, 2, . . . depend on the inlet/outlet duct walls which, for example, may be rigid,

soft or wave-bearing. In the later case the eigenmodes Zj(y), j = 0, 1, 2, . . . will satisfy a

non-Sturm-Liouville eigensystem and the orthogonality relation (OR) will be of the form

(4) together with (5) discussed below. This point is revisited in the section V.

It is assumed that the component lying in the region 0 ≤ x ≤ 2` is of very much more

complicated structure than the inlet/outlet ducts, possibly comprising wave-bearing walls

and layers of absorbent material. The precise details of the component are irrelevant to the

mode-matching procedure, it is required only that there exists an orthogonality relation,

or similar, for the eigenmodes corresponding to the the propagating wave-forms in the

component region. Under such circumstances the velocity potential in the component

region is given by

φ2 =∞∑

j=0

(Bje−sjx + Cje

sjx)Yj(y) (3)

where Bj, Cj are the complex amplitudes of the waves in the component region. The

quantities sj and Yj(y), j = 1, 2, 3 . . . are the wavenumber of the jth mode and the

corresponding eigenfunction. Although details of the eigensystem are not required for the

process described here, it should be noted that the wavenumbers, sj, j = 0, 1, 2, . . . are

defined as the roots of the characteristic equation K(s) = 0 and that the eigenfunctions

must satisfy an OR, or similar, of the form

(Yn, Ym) = δmnEn (4)

where δmn is the usual Kronecker delta function. It is worthwhile pointing out that

such relationships exist not only for Sturm-Liouville eigensystems but also for systems

in which the boundary conditions contain high-order, even derivatives with respect to

the axial direction of the waveguide, that is, x. (A consequence of the latter point is

that K(−s) = K(s).) Such boundary conditions arise if wave-bearing surfaces, such as

a membrane or elastic plate, form part of the component. In such cases the OR will

comprise the usual integral term together with a linear combination of the eigenfunctions

and their derivatives. A typical form for the left hand side of the OR is thus

(Yn, Ym) =∫ d

0Yn(y)Ym(y) dy + P (γ2

n, γ2m)Y ′

n(y0)Y′m(y0) (5)

6

where γn = (−1 − s2n)1/2 and P (u, v) = P (v, u) is a polynomial in the variables u and v,

the coefficients of which depend on the boundary condition describing the wave-bearing

surface which is taken to lie along the line segment y = y0, 0 ≤ x ≤ 2`. A general class of

such expressions was derived by Lawrie and Abrahams10 and these have subsequently been

utilised in a variety of situations.3,11,14 Expression (4) is written as an inner-product. This

formulation is accurate for Sturm-Liouville systems and also those containing membrane

boundaries. For systems with higher-order derivatives present in the boundary conditions

the left hand side of (4) will not be an inner product in the usual sense. This, however, is

immaterial to the following analysis.

Crucial to the process of expressing the mode-matching equations in root-free form is

the knowledge that En is related to the characteristic equation through an expression of

the form

En =Y ′

n(y0)

2sn

d

dsK(s)

∣∣∣∣∣s=sn

(6)

where the prime indicates differentiation with respect to y and y0 is constant, 0 ≤ y0 ≤ h.

The physical significance of the line segment y = y0, 0 ≤ x ≤ 2` depends on the details

of the component but very often represents an interface between fluid and, for example,

a porous material or, as indicated above, the line along which a wave-bearing surface

lies. The precise form of (6) is subject to minor variations, sometimes Yn(y0) appears

rather than its derivative, however, En is invariably proportional to the derivative of the

characteristic equation and inversely proportional to the wavenumber.

Equations (1)–(3) give the form of the wave-field in each region of the duct. The modal

coefficients Aj – Dj, j = 0, 1, 2, . . . are, however, as yet undetermined. Mode-matching is

usually achieved by applying continuity of pressure and normal velocity at the interfaces

between the component and inlet/outlet duct. This process is now demonstrated. For

ease of exposition, the inlet and outlet ducts are assumed to be rigid and plane wave

forcing is be assumed (although neither simplification is a requirement of the solution

method). Thus, the duct modes are given by Zj(y) = cos(jπy/b) and Fj = δj0.

It is convenient to make use of the symmetry of the duct geometry and consider

separately the symmetric and anti-symmetric sub-problems. In both cases only the left

hand side of the system need be considered, and the conditions φ2x(`, y) = 0 or φ2(`, y) =

0 are applied along the line of symmetry for the symmetric and antisymmetric cases

respectively. There are two continuity conditions to be applied at x = 0 and two ORs by

7

which to do this. Which condition is enforced using which OR depends on the geometry.

For h > b continuity of normal velocity is enforced using (4) and continuity of pressure is

enforced using the OR for the standard duct modes, cos(nπy/b)|n = 0, 1, 2, . . .. If h < b

the situation is reversed whilst for h = b there is a choice. It henceforth assumed, without

loss of generality, that h > b. Thus, for the symmetric problem, continuity of pressure

yields the following expression:

ASn =

4

bεn

∞∑

j=0

BSj cosh(sj`)Rnj − δn0. (7)

Here ASn are the reflection coefficients, BS

j are the amplitude of the waves inside the

component for this sub-problem and

Rjm =∫ b

0cos(

jπy

b)Ym(y) dy. (8)

On using (5), it is found that continuity of normal velocity yeilds

BSm =

1

2smEm sinh(sm`)

−iR0m +

∞∑

j=0

ASj ηjRjm − 2Y ′

m(y0)N∑

n=0

aSnγ2n

m

. (9)

It is important to note that for a component with a Sturm-Liouville eigensystem aSn = 0,

n = 0, 1, . . . , N . In contrast, for one involving, for example, an elastic plate these coeffi-

cients are non-zero and must be determined by applying appropriate edge conditions such

as zero-displacement and gradient at the plate edges. It is straightforward to eliminate

Bsm between (9) and (7) to obtain:

ASm =

2

bεm

∞∑

n=0

ASnηnΩnm − 2i

bεm

Ω0m − 4

bεm

N∑

n=0

aSn

∞∑

j=0

coth(sj`)

sjEj

γ2nj Y ′

j (y0)Rmj − δm0 (10)

where

Ωnm =∞∑

j=0

coth(sj`)

sjEj

RmjRmj. (11)

For the anti-symmetric problem a system of equations with identical structure is obtained,

the only difference is that Ωnm is replaced by Λnm which is given by (11) but with coth(sj`)

replaced by tanh(sj`). This replacement also applies to any sums on the right hand side

of (10). Note that the form of Ωnm may vary slightly. For example, if continuity of normal

velocity and pressure are enforced using the opposite choice of OR to that taken here the

wavenumber sj will occur in the numerator of the summand rather than the denominator

and this does occur in the example implemented in the next section. The solution to the

8

full problem is obtained from the symmetric and anti-symmetric sub-problems simply by

noting that An = ASn + AA

n and Dn = ASn − AA

n .

Equation (10) and its anti-symmetric counterpart are of a form that is generic to any

three-part ducting system of the class considered here. The systems may be solved by

truncation and numerical inversion of the matrix. Accurate evaluation of the quantities

Ωjn and Λjn (and any further sums on the right hand side of (10)) for all required frequen-

cies is crucial to this process but, in their current form, this depends on locating sufficient

roots, sj. Note, however, that these roots occur only in within these sums and that the

quantity Em can be eliminated using (6). Thus, Ωnm can be expressed as

Ωnm = 2∞∑

j=0

coth(sj`)

Y ′j (y0)

dds

K(s)|s=sj

RmjRmj. (12)

Once expressed in this form it can be seen that all such sums exhibit two important

features. Firstly, the summands are even functions of sj. This is due to the fact that

the governing equation (Helmholtz’) contains only even powers of both x and y, and all

boundary conditions for the component region are either Sturm-Liouville or contain only

even higher derivatives in x. Secondly, each sum has the form of an infinite sum of the

residues arising from an odd integrand. It is the latter fact that ensures that each sum can

be recast in a form that is independent of the roots, sj, of the characteristic equation for

the component region. All that is required is to choose an appropriate contour integral.

The process is demonstrated in the next section.

III A simple dissipative silencer

In this section the process of expressing Ωjn and Λjn in root-free form is demonstrated for

a two dimensional, three-part ducting system in which the middle component is a simple

dissipative silencer of the type that is commonly found in a typical HVAC ducting system.

Thus, the section of duct lying in the region 0 < x ≤ 2¯ is lined with a porous material

which occupies the space a ≤ y ≤ b, b > a. The porous media is modelled as an equivalent

fluid with complex speed of propagation cp and complex density ρp = Za/cp where Za is

the impedance of the material. Much experimental work has been done on relating cp

and ρp to the bulk acoustic properties of real porous materials and the complex values of

these parameters are known in principle.6

9

A The mode-matching solution

The velocity potential in the silencer region (0 ≤ x ≤ 2`) must satisfy Helmholtz’s equa-

tion with unit wavenumber in the central passage, 0 ≤ y < a and with wavenumber cf/cp

in the porous media a < y ≤ b. At the interface between the porous media and the fluid

(y = a) it is assumed that the pressure and normal velocity are continuous whilst at the

rigid duct walls the normal velocity is zero. The travelling waveforms in a duct of this

type are easily determined by using separation of variables. A typical mode travelling in

the positive x direction has the form φ2n(x, y) = Yn(y)e−snx where

Yn(y) =

Y1n(y), 0 ≤ y < a

Y2n(y), a ≤ y ≤ b(13)

and the wavenumber sn is defined below as a root of (21). The eigenfunctions Yn(y),

n = 0, 1, 2, . . . are the solutions to the eigensystem:

Y ′′1 − γ2Y1 = 0, 0 ≤ y < a; (14)

Y ′1(0) = 0; (15)

Y ′′2 − λ2Y2 = 0, a ≤ y < b; (16)

Y ′2(b) = 0; (17)

Y1(a) = βY2(a); (18)

Y ′1(a) = Y ′

2(a) (19)

where β = ρp/ρ, γ(s) = (−1−s2)1/2 and λ(s) = (Γ2−s2)1/2 with Γ = icf/cp, the branches

being chosen such that γ(0) = +i and λ(0) = +Γ.

It is readily shown that

Yn(y) =

cosh(γny), 0 ≤ y < a

γn sinh(γna)

λn sinh[λnd]cosh[λn(y − b)], a ≤ y ≤ b

. (20)

Here d = b − a, γn = γ(sn), λn = λ(sn) and sn, n = 0, 1, 2, . . . are those roots of the

dispersion relation K(s) = 0 with <(sn) ≥ 0, where

K(s) = cosh(γa) +βγ sinh(γa)

λ sinh[λ(b− a)]cosh[λ(b− a)]. (21)

Note that the roots sn, n = 0, 1, 2, . . . are numbered by increasing real part. Thus, the

mode with wavenumber s0 is the least attenuated.

10

The orthogonality relation8 for this set of eigenfunctions is

∫ a

0Y1nY1m dy + β

∫ b

aY2nY2m dy = δmnEn (22)

where δmn is the usual Kronecker delta and En is given by (6) with y0 = a.

The velocity potentials φj, j = 1, 2, 3 are given by (1)-(2) and (3) with Zn(y) =

cos(nπy/b), n = 0, 1, 2, . . . and ηj = (j2π2/b2 − 1)1/2 with η0 = i, these definitions being

appropriate for the choice of time dependence. Numerical results for both plane wave

forcing and a multi-modal incident field will be presented in the next section. Thus, the

following analysis is carried out under the assumption of multi-modal forcing, but without

stating explicitly the form of the modal amplitudes. The appropriate form for both types

of forcing are stated in section IV. The mode-matching procedure described in section II

yields the following systems of equations:

χn = Fnη1/2n − 2

bεnη1/2n

∞∑

j=0

(χj + Fjη1/2j )

Λjn

η1/2j

(23)

and

ψn = Fnη1/2n − 2

bεnη1/2n

∞∑

j=0

(ψj + Fjη1/2j )

Ωjn

η1/2j

(24)

where χn = (An + Dn)η1/2n ; ψn = (An −Dn)η1/2

n and Rjm is given by (8). Note that, the

explicit form for Rjm for this problem is given in equation (33). For this coupled system

of equations

Λjn =∞∑

m=0

sm

Em

tanh(sm`)RjmRnm ; (25)

and Ωjn is given by (25) with tanh(sm`) replaced by coth(sm`). The coefficients Bn and

Cn, n = 0, 1, 2, . . . are expressed in terms of An and Dn by

Bm + Cm =1

Em

∞∑

j=0

(Aj + Fj)Rjm (26)

and

Bme−2sm` + Cme2sm` =1

Em

∞∑

j=0

DjRmj. (27)

B Recasting the matrix elements

As discussed in section II, expression (25) can be recast in a form whereby the summation

takes place over eigenvalues that can be expressed in closed form. The key is expression

11

(6) which relates the quantity En to the derivative of the dispersion relation. This enables

(25) to be recognised as the sum over a family of poles for a carefully selected integral.

The appropriate integral is

Ijm = limX→∞

∫ iX

−iX

`Υ(s)Lj(s)Lm(s)

coth(s`)ds (28)

where X >> 1 is real. The quantities Υ(s) and Lj(s) are defined by

Υ(s) =s2γ sinh(γa)

`K(s)(29)

and

Lj(s) =Pj(s)

γ2 + (jπ/b)2− Qj(s)

λ2 + (jπ/b)2, (30)

with γ = (−1− s2)1/2 and λ = (Γ2 − s2)1/2, and

Pj(s) = cos(jπa

b) +

jπ

bsin(

jπa

b)

cosh(γa)

γ sinh(γa); (31)

Qj(s) = (−1)j

cos(

jπd

b) +

jπ

bsin(

jπd

b)

cosh(λd)

λ sinh(λd)

. (32)

The choices of Pj(s) and Qj(s) are such that Lj(s) does not have poles when γ2+(jπ/b)2 =

0 or λ2 + (jπ/b)2 = 0 (provided j > 0). Further, although Qj(s) seems to be written in a

slightly cumbersome form, it is this form that ensures that Q(τj) = 0 by inspection (i.e.

without recourse to any trigonometric relations) where τj = (Γ2 + j2π2/b2)1/2. Note that,

the function Lj(s) is related to the quantity Rjn, see (8), by

γn sinh(γna)Lj(sn) = Rjn. (33)

The path of integration in (28) lies along the imaginary axis and is indented to the

left(right) of any poles on the upper(lower) half of the imaginary axis. Therefore, since

the integrand is an odd function of s, Ijm = 0 as |X| → ∞. Note also that the integrand

is 0(s−3) as |s| → ∞. Thus, on deforming the path of integration onto a semi-circular

arc of radius X in the right-hand half plane, the sum of the residues of all the poles

crossed is zero as X → ∞. The integrand has families of poles when: (i) K(s) = 0,

i.e. s = sn; (ii) cosh(s`) = 0, i.e. s = σ1n = (2n + 1)iπ/(2`); (iii) sinh(λd) = 0, i.e.

s = νn = (Γ2 + n2π2/d2)1/2; (iv) γ sinh(γa) = 0, i.e. s = θn = (n2π2/a − 1)1/2 where

d = b − a and, in each case, n = 0, 1, 2, . . .. Note that, as mentioned above, the forms

of Pj(s) and Qj(s) ensure that the quantity Lj(s) has no poles when γ2 + (jπ/b)2 = 0

12

or λ2 + (jπ/b)2 = 0 provided j > 0. The cases j = 0, m 6= 0; m = 0, j 6= 0 and

m = j = 0, however, need separate consideration. For these values of m and j the

families of poles denoted (ii) and (iii) above vanish leaving, in both cases, only the simple

pole corresponding to n = 0.

The first family of poles yields Λjm and, on evaluating all the other pole contributions,

it is found that

1

2Λjm = −

∞∑

n=0

Υ(σ1n)Lj(σ1n)Lm(σ1n) (34)

+(−1)j+m

βd

∞∑

n=0

νnVjn(d)Vmn(d)

εn coth(νn`)+

∞∑

n=0

θnVjn(a)Vmn(a)

aεn coth(θn`)

where

Vmn(x) =

mπb

sin(mπxb

)

[(mπ/b)2 − (nπ/x)2], m 6= n,mx 6= nb

x, m = n = 0

x2(−1)m+n, mx = nb,m 6= 0

(35)

with x real. Note first that the second sum on the right hand side of (34) is of identical

structure to the third sum but with θn replaced by νn and a by d = b− a. Had Qj(s) not

been defined as in (32) the trigonometric terms in the numerator of the second summand

would have contained a rather than d producing an incorrect value when md = nb and

m = j = 0. Note secondly that, for the cases j = 0, m 6= 0; m = 0, j 6= 0 and m = j = 0

the second and third sums of (34) both reduce to one term. This is consistent with the

vanishing of the families of poles denoted by (ii) and (iii) for these values of m and j as

discussed above.

A similar expression for Ωjm can be obtained using the integral (28) but with the term

coth(s`) in the denominator of the integrand replaced with tanh(s`). The expression for

Ωjm is of identical structure to (34) except that every occurence of coth is replaced by

tanh and every occurence of σ1n is replaced by σ2n = nπi/`, n = 0, 1, 2, . . . . Having recast

the matrix elements into a form that is independent of the roots of (21), equations (23)

and (24) many be truncated and solved numerically to determine the amplitudes An and

Dn.

13

C The Power Balance

In the previous section it was shown that it is possible to fully determine the complex

amplitudes An and Dn without solving the complicated dispersion relation K(s) = 0

where K(s) is given by (21). Clearly the incident, reflected and transmitted powers are

independent of the roots of the dispersion relation and can thus easily be determined.

The expressions for these quantities are well known, that for the incident power is

PInc =1

2<

∞∑

m=0

|Fm|2iη∗mεm

, (36)

where ∗ indicates the complex conjugate. The expressions for the reflected and transmitted

powers, Pref and PTrans, are identical but with Fm, m = 0, 1, 2, . . . replaced by Am and

Dm respectively. Note that these expressions are given in the form of power flux per unit

height. Further, Fm, m = 0, 1, 2, . . . are defined such that the incident power as given by

(36) is unity.

It is a trivial matter to deduce that the power absorbed by the layer of lining is

PInc − PRef − PTrans. Such an expression, however, is inadequate in two respects. First,

it does not give any indication of the power flux across each of the three surfaces of

the absorbent lining. Second, it does not complete the power balance and thus, the

opportunity to implement an algorithmic check on the algebra is missed. In fact, it is

straightforward to calculate the power flux across the vertical surfaces at a ≤ y ≤ b,

x = 0 and a ≤ y ≤ b, x = 2` using the non-dimensional flux integral

P =1

b<

−i

∫ b

aφj

(∂φj

∂x

)∗dy

(37)

where j = 1 for the surface at x = 0 and j = 3 for the surface at x = 2`. It is found that

the power absorbed across x = 0 is

Px=0 = <

i

b

∞∑

n=0

∞∑

j=0

(Fn + An)(Fj − Aj)∗η∗j Ijn

, (38)

whilst that absorbed across the surface at x = 2` is

Px=2` = <−

i

b

∞∑

n=0

∞∑

j=0

DnD∗jη∗j Ijn

(39)

where

Ijn =∫ b

acos

(jπy

b

)cos

(nπy

b

)dy (40)

14

which is easily evaluated and independent of the roots sn, n = 0, 1, 2, . . .. The expressions

for Px=0 and Px=2` given above were calculated using φ1 and φ3 respectively and are the

simplest formulae for the power flux across the vertical surfaces. It would, however, have

been equally appropriate to use φ2 (in which case (37) must be multiplied by β = ρp/ρ

to account for the difference in density of the absorbent lining and the acoustic medium).

This yields equivalent expressions in terms of the amplitudes Bn and Cn. It is found that

Px=0 = <

i∞∑

n=0

∞∑

m=0

∆mnG∗nm(0)

λ2m − (λ∗n)2

; (41)

where

∆mn =β

β∗γm sinh(γma) cosh(γ∗na)− γ∗n sinh(γ∗na) cosh(γma) (42)

and

Gmn(x) = sm(Cme2smx −Bme−2smx)(B∗ne−2s∗nx + C∗

ne2s∗nx). (43)

The power flux across the surface at x = 2` is given by (41) with G∗mn(0) replaced by

−G∗mn(`). These expressions depend on the wavenumbers sn and the amplitudes Bn and

Cn, however, they can be recast in a form that is independent of these quantities. The

new forms depend only on An, Dn, Fn and the independent eigenvalues νn, σ1n and σ2n

defined in section III B. Details of the rearrangement are given in the appendix where it

is shown that

Px=0 = <

i

b

∞∑

j=0

∞∑

n=0

[A∗

jAnS1jn − A∗jDnS2jn

] (44)

and

Px=2` = <

i

b

∞∑

j=0

∞∑

n=0

[D∗

jDnS1jn − AnD∗jS2jn

] (45)

where Aj = (Aj + Fj) and

S1jn =∞∑

m=0

(−Υ(σm)Ln(σm)Qj(σm)

λ2(σm) + (jπ/b)2− 2(−1)n+j

βd

νmVnm(d)Vjm(d)

εm tanh(2νm`)

)(46)

and

S2jn =∞∑

m=0

(Υ(σm)Ln(σm)Qj(σm)

(−1)m+1[λ2(σm) + (jπ/b)2]− 2(−1)n+j

βd

νmVnm(d)Vjm(d)

εm sinh(2νm`)

). (47)

Note that, σm = imπ/(2`) which comprises both σ1m and σ2m as defined in section III B.

The power flux across the horizontal surface of the lining can be expressed only in

terms of φ2. The appropriate, non-dimensional, flux integral can be evaluated to obtain

Py=a = <

i

b

∞∑

m=0

∞∑

n=0

cosh(γma)γ∗n sinh(γ∗na)

s2m − (s∗n)2

[G∗nm(`) + Gmn(0)−Gmn(`)−G∗

nm(0)]

.

(48)

15

This expression can also be re-expressed in a form that is independent of Bm, Cm and

sm. The new form depends on An, Dn, Fn and the independent eigenvalues θn, σ1n and

σ2n defined in section III B. Again, details of the rearrangement, which involves some

complicated manipulations, are presented in the appendix. The appropriate form for the

power flux across the horizontal surface of the lining is stated here as

Py=a = <

i

b

∞∑

n=0

∞∑

j=0

[AnD∗

j + DnA∗j

]S3jn −

[DnD∗

j + AnA∗j

]S4jn

(49)

where

S3jn =∞∑

m=0

(Υ(σm)Ln(σm)Pj(σm)

(−1)m+1[γ2(σm) + (jπ/b)2]+

2θmVnm(a)Vjm(a)

aεm sinh(2θm`)

)(50)

and

S4jn =∞∑

m=0

(−Υ(σm)Ln(σm)Pj(σm)

γ2(σm) + (jπ/b)2+

2θmVnm(a)Vjm(a)

aεm tanh(2θm`)

)(51)

with σm = imπ/(2`).

It is not difficult to show that S4jn − S1jn = Λjn + Ωjn/2 and S3jn − S2jn =

Ωjn − Λjn/2. It follows from (44), (45) and (49) that the total power absorbed by the

silencer lining is given by

PAbs = <−i

2b

∞∑

j=0

∞∑

n=0

[A∗jAn + D∗

jDn](Λjn + Ωjn) + [D∗j An + A∗

jDn](Λjn − Ωjn)

.

(52)

This expression has been calculated directly from the flux integrals and is independent of

the reflected and transmitted powers. Thus, the power balance can now be stated as

PRef + PTrans + PAbs = PInc. (53)

where PInc, PRef , PTrans and PAbs are given by (36) and (52) respectively.

For problems involving acoustic transmission in non-dissipative conditions, it is well

established that the systems of equations derived via mode-matching or similar methods

automatically satisfy the power balance regardless of the level of truncation used during

numerical solution.9,14 Further, Warren et. al.14 demonstrate that even if the system of

equations is truncated radically, so that fewer equations are retained than the number

of cut-on modes, the power balance is still satisfied! (Although clearly, in this case, the

distribution of power between the reflected and transmitted components will be incorrect.)

This indicates that, whilst it is necessary that the various components of power should

16

balance this, in itself, is not sufficient to ensure that the numerical solution obtained via

truncation and inversion represents the actual solution to the physical problem under

consideration. The authors are not aware of any equivalent results in the literature for a

dissipative system and, in view of the importance of this issue, it is now shown that the

system of equations, (23) - (25), automatically satisfies the power balance regardless of

level of truncation. It is assumed that the system of equations is truncated to T +1 terms

where T > 0. Thus, on adding and subtracting (23) and (24), it is found that

Amηm = Fmηm − 1

bεm

T∑

j=0

[Aj(Λjm + Ωjm) + Dj(Λjm − Ωjm)] (54)

and

Dmηm = − 1

bεm

T∑

j=0

[Aj(Λjm − Ωjm) + Dj(Λjm + Ωjm)] (55)

where m = 0, 1, 2, . . . , T and Aj = (Aj+Fj). Now the sum of the reflected and transmitted

powers can be written as

PRef + PTrans =1

2

−i

T∑

m=0

[A∗m(Amηm)εm + D∗

m(Dmηm)εm]

(56)

and on using (54) and (55), this becomes

PRef + PTrans =1

2<

−i

(T∑

m=0

A∗mFmηmεm (57)

− 1

b

T∑

m=0

T∑

j=0

[A∗mAj + D∗

mDj](Λjm + Ωjm)

− 1

b

T∑

m=0

T∑

j=0

[A∗mDj + D∗

mAj](Λjm − Ωjm)]

.

Since only the real part is of interest, the first term on the right hand side of (57) can

be replaced with its conjugate. Then, on using (54) to eliminate Am from this term, it is

found that

PRef + PTrans =1

2<

i

T∑

m=0

η∗mF ∗mFmεm (58)

− i

b

T∑

m=0

T∑

j=0

η∗mηm

[F ∗

mAj(Λjm + Ωjm) + DjF∗m(Λjm − Ωjm)

]

+1

2<

i

b

T∑

m=0

T∑

j=0

[A∗mAj + D∗

mDj](Λjm + Ωjm)

+i

b

T∑

m=0

T∑

j=0

[A∗mDj + D∗

mAj](Λjm − Ωjm)]

.

17

On referring to (59) it is seen that Fj = 0 for j ≥ MI where MI is the number of cut-on

modes, whilst ηj is imaginary for j < MI . Thus, without loss of generality the quantity

η∗m/ηm on the right hand side of (58) can be replaced by −1. Then, on collecting together

the terms involving Λjm ± Ωjm, it becomes apparent that the right hand side of (58)

comprises the incident power and that absorbed by the silencer, see (36) and (52), where

both expressions are truncated to T +1 terms. Hence, it is shown that the power balance is

an algebraic identity and in no way guarantees that the numerical solution has converged

to that representing the physical problem.

IV Numerical Results

All the graphical results presented in this section are calculated using the root free expres-

sions given in section III. Two silencer configurations are studied. These are identical in

height (b = 0.6m) and half-length (¯= 1.5m) and are lined with the same absorbent ma-

terial. The difference lies in the depth of the absorbent liner. For silencer 1 the absorbent

layer is thick at 0.45m, thus a = 0.15m. For silencer 2 the depth of the absorbent layer

is much thinner at 0.15m so that a = 0.45m. The absorbent material is characterised

by the regression formulae of Delany and Bazley6,7 with flow resistivity of 8000 rayl/m.

Note, however, that the formulae of Delany and Bazley are known to be invalid at low

frequency and so the semi-empirical correction of Kirby and Cummings6 are then used.

The usual measure of performance for a dissipative silencer is transmission loss: L =

−10 log10(PTrans/PInc) where PTrans and PInc are given by (36). The incident field as

defined by (1) may be either plane wave, in which case the modal amplitudes are given by

Fj = δj0, or multi-modal. Mechel12 suggests that “equal modal energy density” (EMED)

is the most plausible form of multi-modal forcing for this class of system and, under this

assumption, the modal amplitudes are given by

F 2j =

2i

εj∑MI−1

m=0 ηm

, j < MI

0, j ≥ MI

(59)

where εj = 2 for j = 0 and 1 otherwise. Note that, in (59) MI is the number of waves

cut-on in the inlet duct and will depend on b = kb.

Figure 2 shows transmission loss against frequency for silencer 1. Both EMED and

plane wave forcing are shown. At frequencies below 286Hz the two curves overlie as is to

18

be expected since EMED reduces to plane wave forcing below the first cut-on frequency.

Thereafter the transmission loss is slightly higher for plane wave forcing, but the difference

is at most 6 decibels. The results presented in Figure 2 are validated in Table I where, for

seven specified frequencies, the value of the transmission loss obtained using the root-free

method is compared both with that obtained using conventional “rooty” mode-matching

(i.e. solving the characteristic equation in order to obtain sufficient wavenumbers, sm,

by which to accurately evaluate the quantities Ωjn and Λjn) and a finite-element based

point-collocation method.7 All three methods show good agreement.

The power flux across each face of silencer 1, for plane wave and EMED forcing, are

shown in figures 3 and 4 respectively. It is interesting to note that, although the overall

transmission loss is not vastly different between the two types of forcing, the silencer

surface primarily involved in the sound absorption depends on the type of forcing. For

plane wave forcing (figure 3) it is the front face of the lining, i.e. the surface lying along

x = 0, a ≤ y ≤ b, across which the power flux is the greatest, although the flux across

horizontal surface, i.e. that lying along y = a, 0 ≤ x ≤ 2`, is also significant. Whereas

for EMED forcing (figure 4), it is clear that the power flux across the horizontal face of

the silencer is the greatest. In this case, particularly at frequencies greater that 1000Hz,

the energy absorbed across the front face of the silencer is relatively insignificant. For

both forcing mechanisms and indeed both silencer configurations, the flux across rear face

of the silencer, i.e. that lying along x = 2`, a ≤ y ≤ b, is negligible. Indeed as the

frequency tends towards zero, energy actually leaks through this surface which manifests

as a negative flux. The amount of power reflected for the two forcing mechanisms is

similar, although the characteristic spikes at each cut-on frequency are more apparent in

figure 4 (EMED forcing).

Figure 5 shows transmission loss against frequency for silencer 2. Again, both EMED

and plane wave forcing are shown. As for silencer 1, the two curves overlie for frequencies

below 286Hz but, in this case, they also remain very close for frequencies up to 850Hz.

Thereafter, the two curves diverge but the maximum difference is still in the region of

5 decibels. There are two points to be noted. First, at all frequencies, the transmission

loss for silencer 2 is significantly less than that of silencer 1. This is to be expected

since silencer 2 has the thinner lining of the two silencers . Second, for this silencer the

transmission loss is slightly higher for EMED forcing as opposed to plane wave. Again, the

19

results presented in Figure 5 are validated, in Table II, by comparison with the “rooty”

approach and point-collocation.

Figures 6 and 7 show the power flux across the component surfaces of the silencer

for plane wave and EMED forcing respectively. For plane wave forcing both the front

and the horizontal faces of the silencer are proactive i.e. the power flux across them is

significant. It cannot now be said that the front face is the most proactive. This is to be

expected since the silencer lining is comparatively thin and there is, therefore, less sound

incident directly onto the front face than with the thicker lining of silencer 1. For EMED

forcing, however, it is clear that the horizontal surface accounts for the vast majority of

the transmission loss as it does for silencer 1.

V Discussion

It has been demonstrated that a broad class of problem involving the transmission of

sound through a two-dimensional, three-part ducting system can be successfully solved

using analytic mode-matching, but without explicit knowledge of any of the roots of the

characteristic equation for the “middle” region. Here the method was implemented for a

three-part system comprising rigid inlet and outlet ducts with simple silencer sandwiched

between them. The systems of equations obtained via mode-matching were recast, using a

contour integral technique, into a form that is independent of the roots to the characteristic

equation for the silencer region. Robust and accurate root-free expressions by which to

compute all the physical quantities of interest were obtained.

In order to use the method described in this article it is required only that there

exists an appropriate orthogonality relation for the eigenfunctions of the “middle” or

“component” region. For problems in which the inlet/oulet ducts are acoustically hard or

soft the systems of equations obtained via mode-matching can be cast into a form that

involves no root-finding. For situations in which the inlet/oulet ducts are bounded by

wave-bearing surfaces such as a membrane of elastic plate, however, the approach is still of

value. The underlying eigensystems for such inlet/oulet ducts are non-Sturm-Liouville but

are well studied.3,10 Further, although the admissible wavenumbers must be determined

numerically, this task is not usually onerous since the roots to the characteristic equation

are known to lie only on either the real or imaginary axis and can be located with relatively

20

little effort. Thus, for a three-part ducting system comprising inlet/outlet ducts with wave-

bearing boundaries and a middle component of more complicated structure (possibly

comprising both wave-bearing boundaries and layers of porous material) this method

will by-pass root-finding for the middle region thereby significantly reducing the overall

burden of root-finding. Furthermore, it seems likely that this approach can be extended to

three-part ducting systems with circular cylindrical geometry comprising rigid inlet/outlet

ducts. All that is required is that a suitable OR exists for the component region.

A minor disadvantage is that this approach, although highly accurate, is computation-

ally slower that the root finding alternative. This disadvantage may be offset, however,

against the advantage of eliminating the inaccuracies that can arise due to missing roots.

This is of particular importance when plotting physical quantities, such as transmission

loss, against frequency.

A Calculation of the absorbed power

In section III C the power absorbed across each of the three faces of the silencer lining

was given, in terms of the roots sn, n = 0, 1, 2, . . . of the dispersion relation, by (41).

This expression can be recast into forms that depend only on An, Dn, σn = inπ/(2`) and

νn = (Γ2 + n2π2/d2)1/2 and which are, therefore, independent of sn. Consider first the

vertical faces, the first step in recasting (41) is to eliminate Bn and Cn using (26), (27)

and the following expressions

Bn − Cn =coth(2sn`)

En

∞∑

j=0

AjRjn − 1

En sinh(2sn`)

∞∑

j=0

DjRjn;

(A.1)

Bne2sn` − Cne

−2sn` = −coth(2sn`)

En

∞∑

j=0

DjRjn +1

En sinh(2sn`)

∞∑

j=0

AjRjn

(A.2)

where Aj = (Aj+Fj). Due to the similarity in the structures of (26) and (27) and also (A.1)

and (A.2), it is only necessary to derive the root-free expression for (41). The appropriate

result for the power absorbed across the vertical face at x = 2` can then be obtained

by interchanging the coefficients Dj and Aj (and likewise Dq and Aq) throughout. On

eliminating Bn and Cn from (41) and interchanging the orders of summation, it is found

21

that

Px=0 = <−

i

b

∞∑

j=0

∞∑

q=0

(AjA

∗q

∞∑

n=0

s∗nR∗qnT1jn

tanh(2s∗n`)E∗n

− AjD∗q

∞∑

n=0

s∗nR∗qnT1jn

sinh(2s∗n`)E∗n

) (A.3)

where

T1jn =∞∑

m=0

∆mnRjm

[λ2m − (λ∗n)2]Em

=γ∗n sinh(γ∗na)Qj(s

∗n)

[(jπ/b)2 + (λ∗n)2](A.4)

with ∆mn defined by (42). Note that the right hand side of (A.4) was obtained using a

similar contour integral approach to that described in section III B. It follows that

Px=0 = <−

i

b

∞∑

j=0

∞∑

q=0

[AjA∗qS1∗jq − AjD

∗qS2∗jq]

(A.5)

where

S1jq =∞∑

n=0

snγn sinh(γna)RqnQj(sn)

tanh(2sn`)En[(jπ/b)2 + (λn)2](A.6)

and S2jq is identical in structure to S1jq but with the term tanh(2sn`) in the denominator

of the summand replaced by sinh(2sn`). Again, the method described in section III B can

be used to express the quantities S1jq and S2jq in the forms (46) and (47). For S1jq, the

appropriate integral is

J(S1)jq = lim

X→∞

∫ iX

−iX

`Υ(s)Lq(s)Qj(s)

tanh(2s`)[(jπ/b)2 + λ2]ds. (A.7)

and for S2jq the integral is the same except the quantity tanh(2s`) in the denominator

of the integrand is replaced by sinh(2s`). To put (A.5) in exactly the same form as (44),

it is necessary only to replace the summand with its conjugate and multiply the whole

expression by −1. Finally, as mentioned above, the equivalent expression for the power

absorbed by the vertical face at x = 2`, that is Px=2`, is obtained simply by interchanging

the quantities Aj and Dj (likewise Aq and Dq).

Now consider the power absorbed by the horizontal face of the silencer lining,this is

given, in terms of the roots sn, n = 0, 1, 2, . . . of the dispersion relation, by (48). On

interchanging the counters in the second and fourth sums, (48) may be expressed as

Py=a = −<

i

b

∞∑

m=0

∞∑

n=0

cosh(γma)γ∗n sinh(γ∗na)

s2m − (s∗n)2

[Gmn(`)−Gmn(0)] (A.8)

− i

b

∞∑

m=0

∞∑

n=0

cosh(γna)γ∗m sinh(γ∗ma)

s2n − (s∗m)2

[G∗mn(`)−G∗

mn(0)]

where Gmn(x) is defined in (43).

22

It is intended to deal only with the first and third terms of (A.8), i.e. those containing

`. The equivalent results for the second and third terms can be deduced by interchanging

Aj and Dj (likewise Aq and Dq) and changing the sign of the expression. Thus, on using

(27) and (A.2) to re-express the first and third terms of (A.8) in terms of the coefficients

Dj and Aj, and noting that( ∞∑

n=0

γn sinh(γna)Rjn

En(s2n − (s∗m)2)

)∗= −γm sinh(γma)L∗j(sm)

K∗(sm)(A.9)

and∞∑

n=0

cosh(γna)Rjn

En(s2n − (s∗m)2)

= −cosh(γ∗ma)Lj(s∗m)

K(s∗m)+

Pj(s∗m)

(jπ/b)2 + (γ∗m)2(A.10)

where K(s∗m) 6= K∗(sm) and L(s∗m) 6= L∗(sm), it is found that

<

i

b

∞∑

m=0

∞∑

n=0

cosh(γma)γ∗n sinh(γ∗na)Gmn(`)

s2m − (s∗n)2

− cosh(γna)γ∗m sinh(γ∗ma)G∗mn(`)

s2n − (s∗m)2

= <

i

b

∞∑

j=0

∞∑

q=0

[DjA∗qS3∗jq −DjD

∗qS4∗jq]

. (A.11)

Note that, the results given in (A.9) and (A.10) are again proven using contour integration.

Now, on adding in the contributions from the terms in (A.8) that do not include `, it is

found that

Py=a = −<

i

b

∞∑

j=0

∞∑

q=0

[DjA∗q + D∗

qAj]S3∗jq − [DjD∗q + AjA

∗q]S4∗jq

. (A.12)

On taking the complex conjugate of the summand and changing the sign of the whole

expression, this is easily recognised as (49).

Note that, in (A.11), S4jq is given by

S4jq =∞∑

m=0

smγm sinh(γma)RqmPj(sm)

tanh(2sm`)Em[(jπ/b)2 + (γm)2]. (A.13)

and S3jq is identical in structure to S4jq but with tanh(2sn`) replaced by sinh(2sn`).

These sums can be cast in the forms given in (50) and (51) using the the integral approach

described in section III B. The appropriate integrals have a similar form to (A.7).

References

[1] H. Besserer and P.G. Malischewsky, “Mode series expansions at vertical boundaries in

elastic waveguides” Wave Motion, 39, 41-59 (2004).

23

[2] A. Cummings and N, Sormaz, “Acoustic attenuation in dissipative splitter silencers

containing mean fluid flow” J. Sound Vib. 168, 209–227 (1993).

[3] D.V. Evans and R. Porter “Wave scattering by narrow cracks in ice sheets floating on

water of finite depth. ” Journal of fluid Mechanics, 484, 143–165 (2003).

[4] L. Huang “A theoretical study of duct noise control by flexible panels” J. Acoust. Soc.

Am. 106, 1801–1809 (1999).

[5] L. Huang “Modal anaylis of a drumlike silencer” J. Acoust. Soc. Am. 112, 2014–2025

(2002).

[6] R. Kirby and A. Cummings “Prediction of the bulk acoustic properties of fibrous

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25

Table I: Silencer one, EMED forcing: comparison of transmission loss data for the “root-

free”, “rooty” and point-collocation methods.

Silencer 1: EMED forcing

Frequency Non-rooty Rooty Point Collocation

63 16.34688936 16.34478568 16.56900793

125 27.24962896 27.24679555 27.30626097

250 46.21350006 46.21352061 46.18514037

500 53.41436969 53.41410153 53.40450771

1000 50.01804544 50.01620733 50.00904946

2000 19.11579843 19.11573366 19.11350138

4000 8.947613426 8.948377897 8.946284148

Table II: Silencer two, EMED forcing: comparison of transmission loss data for the

“root-free”, “rooty” and point-collocation methods.

Silencer 2: EMED forcing

Frequency Non-rooty Rooty Point Collocation

63 1.828077049 1.829027604 1.85508498

125 6.784509519 6.78685205 6.822066684

250 20.63940465 20.64184964 20.69345116

500 11.40109296 11.40130263 11.4015143

1000 5.355438508 5.355675135 5.356555514

2000 5.125842846 5.125912909 5.126296988

4000 5.504838985 5.504690444 5.504658887

26

Figure 1: Silencer geometry.

0 1000 2000 3000 4000Frequency Hz

10

20

30

40

50

60

70

Transmission

loss

dB

EMED

PW

Key

Figure 2: Transmission loss against frequency for silencer one. Both plane wave and

EMED forcing are depicted.

0 1000 2000 3000 4000Frequency Hz

0

0.2

0.4

0.6

0.8

1

Energy

Ref

B

F

HKey

Figure 3: Proportion of energy reflected and absorbed across each surface of silencer one

for plane wave forcing. The x = 0, y = a and x = 2` surfaces of the silencer are denoted

by F, H and B respectively. Ref indicates the reflected component of power.

27

0 1000 2000 3000 4000Frequency Hz

0

0.2

0.4

0.6

0.8

1

Energy

Ref

B

F

HKey

Figure 4: Proportion of energy reflected and absorbed across each surface of silencer one

for EMED forcing. The x = 0, y = a and x = 2` surfaces of the silencer are denoted by

F, H and B respectively. Ref indicates the reflected component of power.

0 1000 2000 3000 4000Frequency Hz

5

10

15

20

25

30

Transmission

loss

dB

EMED

PW

Key

Figure 5: Transmission loss against frequency for silencer two. Both plane wave and

EMED forcing are depicted.

28

0 1000 2000 3000 4000Frequency Hz

0

0.2

0.4

0.6

0.8

1

Energy

Ref

B

F

HKey

Figure 6: Proportion of energy reflected and absorbed across each surface of silencer two

for plane wave forcing. The x = 0, y = a and x = 2` surfaces of the silencer are denoted

by F, H and B respectively. Ref indicates the reflected component of power.

0 1000 2000 3000 4000Frequency Hz

0

0.2

0.4

0.6

0.8

1

Energy

Ref

B

F

HKey

Figure 7: Proportion of energy reflected and absorbed across each surface of silencer two

for EMED forcing. The x = 0, y = a and x = 2` surfaces of the silencer are denoted by

F, H and B respectively. Ref indicates the reflected component of power.

29